The Bogoliubov inner product (also known as the Duhamel two-point function, Bogolyubov inner product, Bogoliubov scalar product, or Kubo–Mori–Bogoliubov inner product) is a special inner product in the space of operators.
The Bogoliubov inner product appears in quantum statistical mechanics[1][2] and is named after theoretical physicist Nikolay Bogoliubov.
be a self-adjoint operator.
The Bogoliubov inner product of any two operators X and Y is defined as The Bogoliubov inner product satisfies all the axioms of the inner product: it is sesquilinear, positive semidefinite (i.e.,
), and satisfies the symmetry property
In applications to quantum statistical mechanics, the operator
is the Hamiltonian of the quantum system and
With these notations, the Bogoliubov inner product takes the form where
denotes the thermal average with respect to the Hamiltonian
In quantum statistical mechanics, the Bogoliubov inner product appears as the second order term in the expansion of the statistical sum: